Data-driven controller tuning using the correlation approach

The essential ingredients of control design procedures include the acquisition of process knowledge and its efficient integration into the controller. In many practical control applications, a reliable mathematical description of the plant is difficult or impossible to obtain, and the controller has to be designed on the basis of measurements. This thesis proposes a new datadriven method labeled Correlation-based Tuning (CbT). The underlying idea is inspired by the well-known correlation approach in system identification. The controller parameters are tuned iteratively either to decorrelate the closed-loop output error between designed and achieved closed-loop systems with the external reference signal (decorrelation procedure) or to reduce this correlation (correlation reduction). Ideally, the resulting closedloop output error contains only the contribution of the noise and perfect model-following can be achieved. By the very nature of the control design criterion, the controller parameters are asymptotically insensitive to noise. Both theoretical and implementation aspects of CbT are treated. For the decorrelation procedure, a correlation equation is solved using the stochastic approximation method. The iterative procedure converges to the solution of the correlation equation even in the case when an approximate gradient of the closed-loop output error with respect to controller The asymptotic distribution of the resulting controller parameter estimates is analyzed. When perfect decorrelation is not possible, the correlation reduction method can be used. That is, instead of solving the correlation equation, the norm of a cross-correlation function is minimized. A frequency domain analysis of the criterion shows that the algorithm minimizes the two-norm of the difference between the achieved and designed closed-loop systems.With the correlation reduction method, an unbiased estimate of the gradient of the closed-loop output error is necessary to guarantee convergence of the algorithm to a local minimum of the criterion. Furthermore, this criterion can be generalized to allow handling the mixed sensitivity specifications. An extension of this method for the tuning of linear time-invariant multivariable controllers is proposed for both procedures. CbT allows tuning some of the elements of the controller transfer function matrix to satisfy the desired closed-loop performance, while the other elements are tuned to mutually decouple the closed-loop outputs. The tuning of all decouplers and controllers can be made by performing only one experiment per iteration regardless of the number of inputs and outputs since all reference signals can be excited simultaneously. However, due to the fact that decoupling is imposed as a design criterion, simultaneous excitation of all references brings a negative impact on the variance of the estimated controller parameters. In fact, one must choose between low experimental cost (simultaneous excitation) and better accuracy of the estimated parameters (sequential excitation). The CbT algorithm has been tested on numerous simulation examples and implemented experimentally on a magnetic suspension system and the active suspension system benchmark problem proposed for a special issue of European Journal of Control on the design and optimization of restricted-complexity controllers

Modeling of uncertainties in biochemical reactions

Mathematical modeling is an indispensable tool for research and development in biotechnology and bioengineering. The formulation of kinetic models of biochemical networks depends on knowledge of the kinetic properties of the enzymes of the individual reactions. However, kinetic data acquired from experimental observations bring along uncertainties due to various experimental conditions and measurement methods. In this contribution, we propose a novel way to model the uncertainty in the enzyme kinetics and to predict quantitatively the responses of metabolic reactions to the changes in enzyme activities under uncertainty. The proposed methodology accounts explicitly for mechanistic properties of enzymes and physico-chemical and thermodynamic constraints, and is based on formalism from systems theory and metabolic control analysis. We achieve this by observing that kinetic responses of metabolic reactions depend: (i) on the distribution of the enzymes among their free form and all reactive states; (ii) on the equilibrium displacements of the overall reaction and that of the individual enzymatic steps; and (iii) on the net fluxes through the enzyme. Relying on this observation, we develop a novel, efficient Monte Carlo sampling procedure to generate all states within a metabolic reaction that satisfy imposed constrains. Thus we derive the statistics of the expected responses of the metabolic reactions to changes in enzyme levels and activities, in the levels of metabolites, and in the values of the kinetic parameters. We present aspects of the proposed framework through an example of the fundamental three-step reversible enzymatic reaction mechanism. We demonstrate that the equilibrium displacements of the individual enzymatic steps have an important influence on kinetic responses of the enzyme. Furthermore, we derive the conditions that must be satisfied by a reversible three-step enzymatic reaction operating far away from the equilibrium in order to respond to changes in metabolite levels according to the irreversible Michelis-Menten kinetics. The efficient sampling procedure allows easy, scalable, implementation of this methodology to modeling of large-scale biochemical networks

Production of biofuels and biochemicals: in need of an ORACLE

The engineering of cells for the production of fuels and chemicals involves simultaneous optimization of multiple objectives, such as specific productivity, extended substrate range and improved tolerance – all under a great degree of uncertainty. The achievement of these objectives under physiological and process constraints will be impossible without the use of mathematical modeling. However, the limited information and the uncertainty in the available information require new methods for modeling and simulation that will characterize the uncertainty and will quantify, in a statistical sense, the expectations of success of alternative metabolic engineering strategies. We discuss these considerations around the development of a framework for the Optimization and Risk Analysis of Complex Living Entities (ORACLE)

Closed-loop identification of MIMO systems: a new look at identifiability and experiment design

This paper addresses a question raised by a leading expert in the identification of multivariable systems: “Is it necessary to excite all reference signals for the identification of a multivariable system operating in closed loop with a linear time-invariant controller?” On the basis of earlier results on identifiability of closed-loop systems, he conjectured that this was necessary. We show that it is not, on the basis of a careful re-examination of the notions of identifiability and informative experiments for closed-loop systems

Application of the minimum state error variance approach to nonlinear system control

A class of modified state space self-tuning controllers of the minimum state error variance type was considered. A suitable chosen structure of the proposed controller allows the tracking of a time-varying reference input and makes it possible to apply this solution to nonlinear and non-stationary plants. The advantage in using the proposed algorithm for nonlinear system control is demonstrated through its application to aircraft control around a prespecified reference trajectory in the presence of characteristic disturbances. The results show that the proposed controller has good tracking performance and possesses rather good immunity towards disturbances

Correlation-Based Tuning of a Restricted-Complexity Controller for an Active Suspension System

A correlation-based controller tuning method is proposed for the \Design and optimization of restricted-complexity controllers" benchmark problem. The approach originally proposed for model following is applied to solve the disturbance rejection problem. The idea is to tune the controller parameters such that the closed-loop output be uncorrelated with the measured disturbance. Since perfect decorrelation between the closed-loop output and the disturbance is not attainable with a restricted-complexity controller, the cross-correlation of these two signals is minimized. This is done iteratively using stochastic approximation. A frequency analysis of the tuning criterion allows dealing with control speci cations expressed in terms of constraints on the sensitivity functions. Application to the active suspension system of the Automatic Laboratory of Grenoble (LAG) provides a 2nd-order controller that meets the control speci cations to a large extent

Informative data: how to get just sufficiently rich?

Prediction error identification requires that data be informative with respect to the chosen model structure. Whereas sufficient conditions for informative experiments have been available for a long time, there were surprisingly no results of necessary and sufficient nature. With the recent surge of interest in optimal experiment design, it is of interest to know the minimal richness required of the externally applied signal to make the experiment informative. We provide necessary and sufficient conditions on the degree of richness of the applied signal to generate an informative experiment, both in open loop and in closed loop. In a closed-loop setup, where identification can be achieved with no external excitation if the controller is of sufficient degree, our results provide a precisely quantifiable trade-off between controller degree and required degree of external excitation